Power system simulators

ABSTRACT

This invention relates to a power system simulator which enables power system behaviour to be studied rapidly and resonably accurately. The apparatus comprises a network of components interconnected to form an analogue circuit of the system under investigation, the configuration of the network being identical to that of the system and the components employed being such that phase angle changes in busbar voltages are representec by voltages measured between selected points and power changes are represented by currents or rates-of-change of current in the circuit. The analogue circuit may be based on equations linearized with respect to an initial set of conditions; the inertias of the synchronous machines in the system may be represented in the analogue by capacitors, the reactance and damper winding effects of these machines by an auxiliary network of inductors and resistors, the reactances of lines and transformers by inductors alone, and d.c. links either by passive components, resistors, inductors etc. or operational amplifiers.

United States Patent [191 Ainsworth 1 Nov. 27, 1973 POWER SYSTEMSIMULATORS Primary Examiner-Felix D. Gruber [75] Inventor: John DesmondAinsworth, Stafford, AtmmeyiKelth Mlsegades et England [73] Assignee:The English Electric Company [57] ABSTRACT Limited, London, England Thisinvention relates to a power system simulator [22] Filed: Ju|y 11, 1972which enables power system behaviour to be studied Appl. No.: 270,742

Related US. Application Data [30] Foreign Application Priority DataSept. 19, 1969 Great Britain 46, 224/69 [52] US. Cl 235/185, 235/184,235/15l.2l [51] Int. Cl G06g 7/50 [58] Field of Search 235/184, 185,151.21

[56] References Cited UNITED STATES PATENTS 3,675,002 7/1972 Mitsui etal 235/185 2,516.000 7/1950 Harder 235/185 2,491,095 12/1949 Enns235/185 2,323,588 7/1943 Enns 235/185 rapidly and resonably accurately.The apparatus comprises a network of components interconnected to forman analogue circuit of the system under investigation, the configurationof the network being identical to that of the system and the componentsemployed being such that phase angle changes in busbar voltages arerepresented by voltages measured between selected points and powerchanges are represented by currents or rates-of-change of current in thecircuit. The analogue circuit may be based on equations linearized withrespect to an initial set of conditions; the inertias of the synchronousmachines in the system may be represented in the analogue by capacitors,the reactance and damper winding effects of these machines by anauxiliary network of inductors and resistors, the reactances of linesand transformers by inductors alone, and dc links either by passivecomponents, resistors, inductors etc. or operational amplifiers.

10 Claims, 8 Drawing Figures PATENTED V 3,775,603

SHEET 2 [1F 3 FIG? SHEET 3 0F 3 PAIENTEI] NUV 2 71975 1 POWER SYSTEMSIMULATORS This invention relates to apparatus for simulating changes inpower flows and phase angles in a.c. power systems which may, or maynot, include a dc. link, and is a continuation-impart of my US. Pat.application Ser. No. 74540 filed Sept. 18, 1970, now abandoned.

All large a.c. systems including generators interconnected with loads bytransmission lines and transformers, form an oscillatory system in whichoscillations of power flows and phase angles at frequencies of the orderof 0.01 to 3H2 are initiated by any disturbance such as a temporaryshort circuit.

The behaviour of such systems is of considerable interest for planningand design purposes; for example, pole slipping (transient instability)of the generators may occur in extreme cases following a short-circuitor disconnection of a line or generator and, even if pole slipping doesnot occur, the oscillations resulting from the disturbance normally takesome time to die away and this causes disturbance to consumersthroughout the system.

In accordance with this invention, there is provided apparatus forsimulating transient changes with respect to time in power flows, phaseangles, machine speeds, and frequencies in an a.c. power systemembodying synchronous machines caused by disturbances applied to thesystem, including an analogue network of electrical components, and atleast one source of electrical power connectable to said network,wherein the network components are selected and connected such that thedifferential equations relating the currents and voltages in saidnetwork are substantially similar to the linearised differentialequations relating said changes of powers, phase angles, machine speeds,and frequencies in said a.c. power system, said electrical power sourcebeing connected to provide analog functions of defined disturbancesapplied to said a.c. power system, and the apparatus including alsoindicating means connected to said network so as to be responsive toelectrical quantities in the network and'to provide output signals whichare analog functions of said changes in powers, phase angles, machinespeeds, and frequencies in said a.c. power system.

Apparatus in accordance with this invention is inexpensive to constructand yet enables power system behaviour to be studied rapidly andreasonably accurately. It is particularly suitable for studyingoscillatio damping and for optimising control constants.

Various approximations may be made in connection with the systemoperation, particularly with regard to the linearising of the equations,as suggested above, and thus the analog cannot in general be absolutelyprecise. It will be satisfactory for most purposes, however,particularly if the phase angle changes in the system are relativelysmall for any given disturbance. Larger changes will not be accuratelyrepresented, but nevertheless a deduction can readily be made from theswing angles observed as to whether or not pole slipping is likely tooccur. Critical cases may therefore be selected for subsequent study inmore detail by more complex and expensive methods involving the digitalcomputer.

In order that the invention may be fully understood, some embodimentsthereof will now be described with reference to the accompanyingdrawings in which:

FIG. I is a simplified diagram of synchronous generator connected toanother synchronous machine via a busbar;

FIG. 2 shows an equivalent analog circuit of FIG. 1;

FIG. 3 is a diagram of a more comprehensive system; FIG. 4 shows theequivalent analog circuit of FIG. 3;

FIG. 5 shows an accurate analog circuit for a synchronous machine;

FIG. 6 is a diagram of a dc link interconnecting two a.c. systems forwhich an analog circuit may be derived; and

FIG, 7 shows the equivalent analog circuit of FIG. 6;

FIG 8 shbws the operating waveforms on the analog for a particulardisturbance.

Referring now to FIG. 1, there is shown a synchronous generator,depicted as having a constant alternating e.m.f. E at absolute phaseangle 0 with a fixed reactance X connected via a busbar at voltage V/0to another synchronous machine. This latter machine will initially beassumed to be of infinite inertia and possess zero impedance, so thatthe busbar can be regarded as infinite.

With this arrangement, the electrical power P is given y P= EV sin 49/):

= E V- O/X for small values of 0 The shaft of the generator is assumedto be driven at a constant torque T by, eg a steam or water turbine, andin the steady state the phase angle 0 will be at a value such that theoutput power P is substantially equal to the mechanical input power. 7

Thus, for speed close to the rated speed, expressing quantities in perunit, P T,,,.

If 0 is difi'erent from this particular value mentioned there is a netaccelerating or retarding power (or torque) acting on the machineinertia, such that where H is the inertia constant of the machine and Wis the rated angular frequency.

Considering the above equations now in terms of changes in power andphase angle from initial stead conditions then EV 1 AP: A0=( l5 A0 (1and 2H/W a (A0)/dt AP from which 1 s A0= AP (2) where s d/dt Referringnow to the circuit shown in FIG. 2, consisting of an inductor L and acapacitor C where v is the voltage across the capacitor and i is thecurrent, this circuit may be described by the differential equationsdi/dt= l/L v and s v= l/C-di/dt...

Thus, from a comparison between equations l and (3) and (2) and (4) itwill be seen that the circuit of FIG. 2 can be considered as an analogof the power system shown in FlG. 1, if the following conditions obtaindi/d!=ABAP.

v=BA6.

C=A(2H/W,,)

L=(l/A) (X/EV) the constants A and B having been introduced merely asscale factors to obtain convenient component values in the mode] oranalog.

Thus assuming the e.m.f. E and the busbar voltage V in the real systemto be fixed, then on the analog di/dt is proportional to a change inreal system power, analog voltage v is proportional to change in realsystem absolute phase angle or shaft angle, analog capacitance C isproportional to the real machine inertia and the analog inductor L isproportional to real machine reactance.

Constant loads (other than synchronous motors) are of no concern and arenot represented in the analog since, on the assumption that the ac.voltage is substantially constant, their power does not changeappreciably.

Because of the linear approximation made above (6 sin6) it can also beassumed that an additional series reactance X, e.g. line reactance,having voltages V, and V at its ends, may be represented in the analogby an inductor.

provided again that the voltages are assumed constant.

in general, it can be shown that the method may be applied to a powersystem having any number of machines interconnected by any number oflines and transformers, applying the rules given in equations (5) to (9)to each component in the system.

One example showing a network of three generators 1, 2, 3 and lines 4,5, 6 together with the associated loads 7, 8, 9 is shown in FIG. 3.

In the corresponding analogue circuit in FIG. 4 the inertias of thethree machines are represented by capacitors C C the machine reactancesX, X by inductors L L machine rotor (and e.m.f.) angle changes byvoltages v 1 v busbar phase angle changes by voltages v v linereactances X, X by inductors Ll L and the power change A P in eachmachine and line by the corresponding model di/d! (equ. 5) which can beobserved as the proportional voltage across an appropriate inductorthrough which the relevant current flows, as by means of indicatingmeans such as an oscilloscope 0 shown, for example, across inductor Ll.It will be seen from equation 6 that voltages across the capacitorsrepresent changes of absolute phase angle and indicating means of anyconvenient kind. such as one or more cathode ray oscilloscopes, areprovided as at M in accordance with the invention for monitoring andindicating these voltages or currents under proper circuit connections.Voltages elsewhere in the analog circuit represent changes of absoluteelectrical phase of voltage at corresponding points in the real system,and can similarly be indicated on a cathode ray oscilloscope.

The manner in which the synchronous machines have been represented aboveneglects the effect of damper windings in these machines, and althoughthis is often adequate for studying the effects of the first peakoccurring it is not adequate for studying damping effects. As a result amore comprehensive representation of the machine is sometimes desirable.

The analog circuit shown in FIG. 5 is suitable for this purpose, thecapacitor C being the same as that shown in FIG. 1, whilst the inductorL and the inductors L L and associated series resistors R R, arecalculated from the basic machine parameters and from the initialloading conditions. For a single machine connected to infinite busbarsthe component values can be shown to be given by:

where V, terminal voltage p.u.

8,, initial rotor angle Q initial reactive power and Xd, X41, X X X TTa, Ta T T have the normal meanings of synchronous machine terrninologyin per unit.

Since R and L are representative of field parame ters, if field voltageis changed by A V this can be represented on the analog by a suitablevoltage source This can conveniently be arranged in practice using anormal electronic amplifier, which can incorporate the time functions of(17).

The input of this amplifier is then an equivalent injection point forfield voltage changes and can be used to inject a small disturbance (asoften done for test purposes on a real system) and also for adding fieldcontrol signals derived from the equivalents of machine power or anglechanges as above. e

The values of L L above are clearly functions of the machine reactancesand initial working conditions. The values of R R further depend onfield and damper resistances. The damping effects of the latter can beshown to be removed if required by making R and R tend to infinity and Rtend to zero, when the resistance/inductance network in FIG. reduces toa single equivalent inductor equal to L and L in parallel, which can beshown to be the same as the single inductor L of FIG. 2.

Where the machine is connected to a finite a.c. system these analogcomponent values are an approximation, but are sufficiently accurate fornormal engineering purposes.

The capacitor C is the same as before from equation 7, and the voltageacross it represents change of absolute shaft angle from equation 6 asbefore. Voltages elsewhere in the analogue represent change of absoluteelectrical phase of the voltage at the corresponding point in the realsystem. Change of machine shaft speed may be derived using therelationship Since the voltage v is directly accessible, it follows fromequation 18 that a signal proportional to the frequency change Af may beobtained from the output of a normal differentiating amplifier DE havingits input derived from the voltage v.

The real system disturbances, e.g. opening and closing circuit-breakers,and short circuits may readily be simulated in the analog.

In particular, the opening of a circuit-breaker to disconnect a line inthe system e.g. S, in FIG. 3 has an effect on phase angles equivalentto:

a. changing the line reactance to infinity, and

b. inserting a new power flow, equal but opposite to that originallyflowing in the line, into each of the two busbars to which the line wasconnected.

On the analog these effects may be obtained as shown in FIG. 4 by a.open-circuiting the point corresponding to the position of thecircuit-breaker by a switch S and b. suddenly injecting ramp-functionsof current equal in magnitude but opposite in sign into the points onthe analog corresponding to the two busbars by sources 25,26, therate-of-change of current (di/dt) being calculated from equation 5,where AP is the line power previously existing. Conventional'electronicwaveform generators and current amplifiers may conveniently be employedas such sources.

The re-c1osing of thecircuit-breaker (if required) is simulated by theconverse steps, that is "by stopping the current ramps at the currentthey may have attained at the particular time, and by re-connectingtogether the points corresponding to the position of the circuitbreakerby switch S The switching action S on the analog may be readily obtainedby any well-known form of electronic switch, e.g. a transistor,controlled by a suitable electronic pulse generator.

After applying the appropriate functions of current corresponding to adesired circuit-breaker opening (followed by re-closing if desired) andcompleting the time desired for observation of the effects on thesystem, it is not important in principle what happens to the currentsinjected into the analog system; however, it is convenient in practiceto restore the currents to zero immediately after this time particularlyif repetitive operation of the analog is required.

With regard to the simulation of a' short circuit,'this produces zerovoltage at the fault point on the real system, e.g. on a line, the powertransfer falling to zero; for present purposes this produces effectsidentical to that of simply opening circuitbreakers in the line, and maybe simulated as above.

A simulation of a load change AP at a particular .point e.g. by opening,or closing circuit breaker S in FIG. 3 may be carried out by arranginga current source such as a suitable electronic amplifier 24 in FIG. 4,to inject a current i at the corresponding point on the analog, where iis related to AP from equation 5 by i=A B I APdt=AB/sAP...

where s is the operator d/dt For example, a suddenly applied load of APfrom equation 19 will require a ramp function of current of rate ABAPsuddenly applied to the analog.

The appropriate measurement on the analog may be made by an oscilloscopefrom which the waveforms at various points may be observed. Therepresentations will give phase angle changes and rotor angle changesdirectly in terms of oscilloscope voltage, from equation 6. Powerchanges may be obtained from current derivative using equation 5, butare more conveniently observed as the proportional voltage across aninductor through which the relevant current flows.

If any d.c. transmission links are connected in the a.c. systems theyare not represented in the analog if they are operated at constantpower.

However, if the d.c. power is controlled in response to signals derivedfrom one of the a.c. systems, e.g. as shown in FIG. 6, then the efi'ectof the d.c. link may be introduced into the analog.

In particular, FIG. 6 shows a system in which an invertor 14 draws powerfrom an a.c. system 15 via a rectifier l6, and supplies power P into areceiving system 17 via a busbar 18. The d.c. link is controlled toprovide the power P substantially equal to an order signal (P order)derived from measuring apparatus 19 which derives a signal proportionalto a change function of absolute phase angle of the voltage on busbar18, e.g. apparatus as described in my copending U.S. Pat. applicationSer: No. 36156 filed May 1 l 1970 now U.S. Pat. No. 3,668,413.

Accordingly,

P order setttnn 6 where P is a fixed signal f(s) is a time function 6 isabsolute phase angle of the busbar voltage and s is the diffentialoperator d/dt as before.

The function f(s) is assumed to be a linear operational function atleast for small changes, and therefore will in general consist of theratio of two polynomials in s, containing only powers of s, andconstants.

It can be shown that the effect of the d.c. link on the ac. receivingsystem 17 is equivalent in the analog to a shunt impedance As anexample, iff(s) Ks where K is a constant, then Z l/AK which correspondsto a pure resistor.

Alternatively, if

fls) KsT/l sT where K and T are constants,

then

Z l/AKT+ s/AK which is a pure resistor l/AKT in series with an inductorl/AK.

Analogs of more complex functions may be obtained although in some casesthese cannot be realised by passive components alone. For example, ananalog circuit of the scheme shown in FIG. 6 is shown in FIG. 7 in whichan operational amplifier is employed.

In this latter Figure an operational voltage amplifier 21 delivers avoltage input to an amplifier 22 which in turn delivers a current outputto a terminal 23 connected to the appropriate point in the overallanalog circuit of the whole scheme, this output being additionally fedback as the input to amplifier 21.

The operational amplifier 21 has a transfer function given by where K isa constant.

This method may readily be extended to the case where control signalsare derived from each end of the dc link, instead of the one end shown,and where the ac. systems at both ends are to be studied.

In practice, it may be convenient to operate the analog faster than realtime, that is to say, faster than the simulated change in the powersystem represented, as is often done in normal analog computingpractice. As an example, for a time scale speed-up factor of 1,000, thefollowing changes are required:

a. After calculating analog components in real time as above, allinductance and capacitance values are decreased by 1,000 times,resistors being left unchanged. This also applies to components directlyassociated with operational amplifier 21.

b. The rate of change of current in any applied ramp function of currentis increased by 1,000 times.

c. The relative timing of analog disturbances (e.g. time betweentripping and re-closing of a circuit breaker) is reduced by 1,000 times.

d. The time scale of voltages and currents observed on the analog isincreased by a factor of 1,000 to obtain equivalent real time scale.

Repetitive operation of the analog may be obtained by restoring allinputs to their initial values after the desired observation time,waiting a further time for transients to die away, then repeating theinput disturbances, and so on. It is a substantial advantage in thiscase to operate faster than real time as above, since with asufficiently high repetition frequency the traces observed on acathode-ray oscilloscope appear to be stationary, and the results ofexperimental changes made to parameters by changing analog componentvalues can be observed almost instantly.

As an example, FIG. 8 shows waveforms relevant to repetitive operationof the analog for the case of an additional load switched on for afinite time T, and then removed, as shown at (a). This corresponds to aninjected current waveform applied to the analogue circuit as shown in(b), consisting of a ramp stopping at time T Assuming observations areonly required up to a time T then at any subsequent time T the injectedcurrent may be restored to zero in preparation for the next transient tobe applied at time T Waveform (c) is of the voltage at some typicalpoint in the analog, representing phase angle change caused by thetransient for example, across the capacitor C l of FIG. 4. At time T theresetting of injected current causes an unavoidable further transient inthe analog. This is outside the time of interest and need not beinspected by oscilloscope, but time T (i.e., repetition period ofoperation) should be sufficiently great that all transients have diedaway by time t= T.,. For a typical a.c. system, if T is say 1 second,then observation time T might be say 10 seconds, T; can be say 12seconds, and T say 20 seconds. The above description was for an analogoperation in real time, but if a time scale speed-up of 1,000 times isused on the analog these correspond to analog times of t l0ms, t l2msand t 20 ms, where the letter I indicates a time on the analogcorresponding to a real system time T, with appropriate subscripts ineach case. Repetition period on the analog is then I, 20 ms, andrepetition frequency 50 per second.

Thus the steps required in practice in order to put the invention intoeffect may be summarised as follows:

a. Determine the single-line equivalent series reactance diagram of thepower system, on a per unit basis.

b. Determine the required initial steady state load flow, includingpower and phase angle at all points.

c. Set up an analog network of inductors in the same topologicalarrangement as the single-line reactance diagram, with values calculatedaccording to equation 9.

d. Set up an analog network corresponding to each machine node similarto FIG. 5 with values calculated from equations 7 and 10 to 16 torepresent each synchronous machine. The simpler arrangement of FIG. 2consisting on one inductor and one capacitor may be used if dampingeffects are not required, or for machines remote from a disturbance.

e. Excite the analog from a suitable electronic waveform generator andelectronic amplifiers to represent a desired disturbance as describedabove. Repetitive operation may be used, as for example as shown in FIG.8.

f. Observe voltage and current waveforms at required points in theanalog by cathode-ray oscilloscope, interpreting voltage as phase angleor machine rotor angle change, rate ofchange of voltage as frequency orspeed change, and rate of change of current as power change. Theconstants of proportionality in these observations are set forth inequation 6 for phase angle or rotor angle, in equation 18 for frequencyor speed change, and in equation for power change.

I claim:

1. Apparatus for simulating transient changes with respect to time inpower flows, phase angles, machine speeds and frequencies in an a.c.power system embodying synchronous machines, including an analog networkof electrical components, and at least one source of electrical currentconnectable to the network, wherein the network components are selectedand connected such that the differential equations relating the currentsand voltages in said network are substantially similar to the linearizeddifferential equations relating small changes of powers, phase angles,machine speeds, and frequencies in said a.c. power system, saidelectrical current source being connected to provide analog functions ofdefined disturbances applied to said a.c. power system, and theapparatus including indicating means connected to the network so as tobe responsive to electrical quantities in the network and to provideoutput signals which are analog functions of changes in powers, phaseangles, machine speeds, and differentiator means to generate a signalresponsive to frequencies in said a.c. power system.

2. Apparatus as claimed in claim 1 in which said analogue networkincludes a first group of components representing the equivalent seriesreactances of lines and transformers in said a.c. power system, and asecond group of components representing each synchronous machine in saidsystem.

3. Apparatus as claimed in claim 2 in which said first group ofcomponents comprises inductors proportional to the per unit value ofeach reactance in the power system and connected in a similartopological configuration.

4. Apparatus as claimed in claim 2 in which a said second group ofcomponents comprises capacitor proportional to the inertia of themachines, inductor means of values which are functions of the per unitreactances of the machines, and resistor means which are functions ofmachine damper winding and field winding resistances.

5. Apparatus as claimed in claim 1 wherein said current source providesan analog of a defined load power change and which is connected to theanalog network at a nodal point corresponding to that of the realsystem.

6. Apparatus as claimed in claim 1 which further includes switch meansfor simulating a disturbance consisting of the opening of a line in thepower system connected between the corresponding two nodes on the analognetwork, and wherein current sources connected to each said node on theanalog network, each capable of producing a current of a value which isa function of the power previously existing in the line.

7. Apparatus according to claim 1 in which said indicating means is acathode-ray oscilloscope.

8. Apparatus according to claim 1 in which said network includescomponents forming an analog of a high voltage direct current powertransmission link in said a.c. power system, said components includingmeans providing a representation of the effect of the control of thepower of said hv-dc link from a function of frequency, phase angle, orpower in said a.c. power system.

9. Apparatus according to claim 8 in which said components forming ananalog of an hv-dc link are passive components.

10. Apparatus according to claim 8 in which said components forming ananalog of an hv-dc link include one or more electronic amplifiers.

1. Apparatus for simulating transient changes with respect to time inpower flows, phase angles, machine speeds and frequencies in an a.c.power system embodying synchronous machines, including an analog networkof electrical components, and at least one source of electrical currentconnectable to the network, wherein the network components are selectedand connected such that the differential equations relating the currentsand voltages in said network are substantially similar to the linearizeddifferential equations relating small changes of poweRs, phase angles,machine speeds, and frequencies in said a.c. power system, saidelectrical current source being connected to provide analog functions ofdefined disturbances applied to said a.c. power system, and theapparatus including indicating means connected to the network so as tobe responsive to electrical quantities in the network and to provideoutput signals which are analog functions of changes in powers, phaseangles, machine speeds, and differentiator means to generate a signalresponsive to frequencies in said a.c. power system.
 2. Apparatus asclaimed in claim 1 in which said analogue network includes a first groupof components representing the equivalent series reactances of lines andtransformers in said a.c. power system, and a second group of componentsrepresenting each synchronous machine in said system.
 3. Apparatus asclaimed in claim 2 in which said first group of components comprisesinductors proportional to the per unit value of each reactance in thepower system and connected in a similar topological configuration. 4.Apparatus as claimed in claim 2 in which a said second group ofcomponents comprises capacitor proportional to the inertia of themachines, inductor means of values which are functions of the per unitreactances of the machines, and resistor means which are functions ofmachine damper winding and field winding resistances.
 5. Apparatus asclaimed in claim 1 wherein said current source provides an analog of adefined load power change and which is connected to the analog networkat a nodal point corresponding to that of the real system.
 6. Apparatusas claimed in claim 1 which further includes switch means for simulatinga disturbance consisting of the opening of a line in the power systemconnected between the corresponding two nodes on the analog network, andwherein current sources connected to each said node on the analognetwork, each capable of producing a current of a value which is afunction of the power previously existing in the line.
 7. Apparatusaccording to claim 1 in which said indicating means is a cathode-rayoscilloscope.
 8. Apparatus according to claim 1 in which said networkincludes components forming an analog of a high voltage direct currentpower transmission link in said a.c. power system, said componentsincluding means providing a representation of the effect of the controlof the power of said hv-dc link from a function of frequency, phaseangle, or power in said a.c. power system.
 9. Apparatus according toclaim 8 in which said components forming an analog of an hv-dc link arepassive components.
 10. Apparatus according to claim 8 in which saidcomponents forming an analog of an hv-dc link include one or moreelectronic amplifiers.